LMFDB - Embedded newform 1216.3.g.e.417.35

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,3,Mod(417,1216)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1216.417");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.g (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			417.35
Character	$\chi$	$=$	1216.417
Dual form			1216.3.g.e.417.34

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.88498 q^{3} -4.10261i q^{5} -9.53824 q^{7} -0.676870 q^{9} +O(q^{10})$	$q+2.88498 q^{3} -4.10261i q^{5} -9.53824 q^{7} -0.676870 q^{9} +13.4664i q^{11} +24.6090 q^{13} -11.8360i q^{15} -3.18711 q^{17} +(5.97352 + 18.0365i) q^{19} -27.5177 q^{21} +13.9757 q^{23} +8.16859 q^{25} -27.9176 q^{27} +37.6700 q^{29} -26.8415i q^{31} +38.8503i q^{33} +39.1317i q^{35} -7.92032 q^{37} +70.9966 q^{39} +24.3998i q^{41} -44.2032i q^{43} +2.77693i q^{45} -2.67164 q^{47} +41.9780 q^{49} -9.19475 q^{51} +58.2044 q^{53} +55.2473 q^{55} +(17.2335 + 52.0351i) q^{57} +99.5738 q^{59} -55.1338i q^{61} +6.45614 q^{63} -100.961i q^{65} +60.1141 q^{67} +40.3197 q^{69} +65.8927i q^{71} +115.370 q^{73} +23.5663 q^{75} -128.446i q^{77} +89.0501i q^{79} -74.4500 q^{81} +29.1508i q^{83} +13.0755i q^{85} +108.677 q^{87} -147.108i q^{89} -234.727 q^{91} -77.4374i q^{93} +(73.9969 - 24.5070i) q^{95} +170.677i q^{97} -9.11499i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 264 q^{9}+O(q^{10}) $ 48 * q + 264 * q^9	$ 48 q + 264 q^{9} + 24 q^{17} - 160 q^{25} + 328 q^{49} + 8 q^{57} - 472 q^{73} - 736 q^{81}+O(q^{100}) $ 48 * q + 264 * q^9 + 24 * q^17 - 160 * q^25 + 328 * q^49 + 8 * q^57 - 472 * q^73 - 736 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	2.88498			0.961661			0.480831	−	0.876814i	$-0.340335\pi$
$3$	2.88498			0.961661			0.480831	+	0.876814i	$0.340335\pi$
$4$		0			0
$5$		−	4.10261i		−	0.820522i	−0.911968	−	0.410261i	$-0.865438\pi$
$5$		−	4.10261i		−	0.820522i	0.911968	−	0.410261i	$-0.134562\pi$
$6$		0			0
$7$	−9.53824			−1.36261			−0.681303	−	0.732002i	$-0.738586\pi$
$7$	−9.53824			−1.36261			−0.681303	+	0.732002i	$0.738586\pi$
$8$		0			0
$9$	−0.676870			−0.0752077
$10$		0			0
$11$			13.4664i			1.22422i	0.790774	+	0.612108i	$0.209678\pi$
$11$			13.4664i			1.22422i	−0.790774	+	0.612108i	$0.790322\pi$
$12$		0			0
$13$	24.6090			1.89300			0.946501	−	0.322701i	$-0.104591\pi$
$13$	24.6090			1.89300			0.946501	+	0.322701i	$0.104591\pi$
$14$		0			0
$15$		−	11.8360i		−	0.789064i
$16$		0			0
$17$	−3.18711			−0.187477			−0.0937384	−	0.995597i	$-0.529882\pi$
$17$	−3.18711			−0.187477			−0.0937384	+	0.995597i	$0.529882\pi$
$18$		0			0
$19$	5.97352	+	18.0365i	0.314396	+	0.949292i
$19$	5.97352	+	18.0365i	0.314396	+	0.949292i
$20$		0			0
$21$	−27.5177			−1.31036
$22$		0			0
$23$	13.9757			0.607640			0.303820	−	0.952729i	$-0.401738\pi$
$23$	13.9757			0.607640			0.303820	+	0.952729i	$0.401738\pi$
$24$		0			0
$25$	8.16859			0.326744
$26$		0			0
$27$	−27.9176			−1.03399
$28$		0			0
$29$	37.6700			1.29896			0.649482	−	0.760377i	$-0.274986\pi$
$29$	37.6700			1.29896			0.649482	+	0.760377i	$0.274986\pi$
$30$		0			0
$31$		−	26.8415i		−	0.865856i	−0.901429	−	0.432928i	$-0.857480\pi$
$31$		−	26.8415i		−	0.865856i	0.901429	−	0.432928i	$-0.142520\pi$
$32$		0			0
$33$			38.8503i			1.17728i
$34$		0			0
$35$			39.1317i			1.11805i
$36$		0			0
$37$	−7.92032			−0.214063			−0.107031	−	0.994256i	$-0.534135\pi$
$37$	−7.92032			−0.214063			−0.107031	+	0.994256i	$0.534135\pi$
$38$		0			0
$39$	70.9966			1.82043
$40$		0			0
$41$			24.3998i			0.595118i	0.954703	+	0.297559i	$0.0961724\pi$
$41$			24.3998i			0.595118i	−0.954703	+	0.297559i	$0.903828\pi$
$42$		0			0
$43$		−	44.2032i		−	1.02798i	−0.857796	−	0.513991i	$-0.828167\pi$
$43$		−	44.2032i		−	1.02798i	0.857796	−	0.513991i	$-0.171833\pi$
$44$		0			0
$45$			2.77693i			0.0617096i
$46$		0			0
$47$	−2.67164			−0.0568435			−0.0284218	−	0.999596i	$-0.509048\pi$
$47$	−2.67164			−0.0568435			−0.0284218	+	0.999596i	$0.509048\pi$
$48$		0			0
$49$	41.9780			0.856694
$50$		0			0
$51$	−9.19475			−0.180289
$52$		0			0
$53$	58.2044			1.09820			0.549098	−	0.835758i	$-0.314971\pi$
$53$	58.2044			1.09820			0.549098	+	0.835758i	$0.314971\pi$
$54$		0			0
$55$	55.2473			1.00450
$56$		0			0
$57$	17.2335	+	52.0351i	0.302342	+	0.912897i
$58$		0			0
$59$	99.5738			1.68769			0.843846	−	0.536585i	$-0.180286\pi$
$59$	99.5738			1.68769			0.843846	+	0.536585i	$0.180286\pi$
$60$		0			0
$61$		−	55.1338i		−	0.903832i	−0.892060	−	0.451916i	$-0.850741\pi$
$61$		−	55.1338i		−	0.903832i	0.892060	−	0.451916i	$-0.149259\pi$
$62$		0			0
$63$	6.45614			0.102478
$64$		0			0
$65$		−	100.961i		−	1.55325i
$66$		0			0
$67$	60.1141			0.897225			0.448612	−	0.893726i	$-0.351918\pi$
$67$	60.1141			0.897225			0.448612	+	0.893726i	$0.351918\pi$
$68$		0			0
$69$	40.3197			0.584344
$70$		0			0
$71$			65.8927i			0.928067i	0.885818	+	0.464033i	$0.153598\pi$
$71$			65.8927i			0.928067i	−0.885818	+	0.464033i	$0.846402\pi$
$72$		0			0
$73$	115.370			1.58041			0.790207	−	0.612841i	$-0.209973\pi$
$73$	115.370			1.58041			0.790207	+	0.612841i	$0.209973\pi$
$74$		0			0
$75$	23.5663			0.314217
$76$		0			0
$77$		−	128.446i		−	1.66812i
$78$		0			0
$79$			89.0501i			1.12722i	0.826042	+	0.563608i	$0.190587\pi$
$79$			89.0501i			1.12722i	−0.826042	+	0.563608i	$0.809413\pi$
$80$		0			0
$81$	−74.4500			−0.919136
$82$		0			0
$83$			29.1508i			0.351214i	0.984460	+	0.175607i	$0.0561888\pi$
$83$			29.1508i			0.351214i	−0.984460	+	0.175607i	$0.943811\pi$
$84$		0			0
$85$			13.0755i			0.153829i
$86$		0			0
$87$	108.677			1.24916
$88$		0			0
$89$		−	147.108i		−	1.65290i	−0.563011	−	0.826450i	$-0.690357\pi$
$89$		−	147.108i		−	1.65290i	0.563011	−	0.826450i	$-0.309643\pi$
$90$		0			0
$91$	−234.727			−2.57942
$92$		0			0
$93$		−	77.4374i		−	0.832660i
$94$		0			0
$95$	73.9969	−	24.5070i	0.778915	−	0.257968i
$96$		0			0
$97$			170.677i			1.75956i	0.475383	+	0.879779i	$0.342310\pi$
$97$			170.677i			1.75956i	−0.475383	+	0.879779i	$0.657690\pi$
$98$		0			0
$99$		−	9.11499i		−	0.0920706i
$100$		0			0
$101$		−	99.4718i		−	0.984869i	−0.870349	−	0.492435i	$-0.836107\pi$
$101$		−	99.4718i		−	0.984869i	0.870349	−	0.492435i	$-0.163893\pi$
$102$		0			0
$103$		−	46.6278i		−	0.452698i	−0.974046	−	0.226349i	$-0.927321\pi$
$103$		−	46.6278i		−	0.452698i	0.974046	−	0.226349i	$-0.0726789\pi$
$104$		0			0
$105$			112.894i			1.07518i
$106$		0			0
$107$	−137.083			−1.28115			−0.640576	−	0.767895i	$-0.721304\pi$
$107$	−137.083			−1.28115			−0.640576	+	0.767895i	$0.721304\pi$
$108$		0			0
$109$	−79.6605			−0.730831			−0.365415	−	0.930845i	$-0.619073\pi$
$109$	−79.6605			−0.730831			−0.365415	+	0.930845i	$0.619073\pi$
$110$		0			0
$111$	−22.8500			−0.205856
$112$		0			0
$113$			3.18887i			0.0282201i	0.999900	+	0.0141100i	$0.00449151\pi$
$113$			3.18887i			0.0282201i	−0.999900	+	0.0141100i	$0.995508\pi$
$114$		0			0
$115$		−	57.3369i		−	0.498582i
$116$		0			0
$117$	−16.6571			−0.142368
$118$		0			0
$119$	30.3994			0.255457
$120$		0			0
$121$	−60.3434			−0.498706
$122$		0			0
$123$			70.3931i			0.572301i
$124$		0			0
$125$		−	136.078i		−	1.08862i
$126$		0			0
$127$		−	87.1184i		−	0.685972i	−0.939341	−	0.342986i	$-0.888562\pi$
$127$		−	87.1184i		−	0.685972i	0.939341	−	0.342986i	$-0.111438\pi$
$128$		0			0
$129$		−	127.525i		−	0.988570i
$130$		0			0
$131$			258.575i			1.97385i	0.161170	+	0.986927i	$0.448473\pi$
$131$			258.575i			1.97385i	−0.161170	+	0.986927i	$0.551527\pi$
$132$		0			0
$133$	−56.9768	−	172.037i	−0.428397	−	1.29351i
$134$		0			0
$135$			114.535i			0.848408i
$136$		0			0
$137$	59.4573			0.433995			0.216997	−	0.976172i	$-0.430374\pi$
$137$	59.4573			0.433995			0.216997	+	0.976172i	$0.430374\pi$
$138$		0			0
$139$			204.541i			1.47152i	0.677242	+	0.735760i	$0.263175\pi$
$139$			204.541i			1.47152i	−0.677242	+	0.735760i	$0.736825\pi$
$140$		0			0
$141$	−7.70765			−0.0546642
$142$		0			0
$143$			331.395i			2.31744i
$144$		0			0
$145$		−	154.545i		−	1.06583i
$146$		0			0
$147$	121.106			0.823849
$148$		0			0
$149$			235.042i			1.57746i	0.614738	+	0.788731i	$0.289262\pi$
$149$			235.042i			1.57746i	−0.614738	+	0.788731i	$0.710738\pi$
$150$		0			0
$151$			198.170i			1.31238i	0.754595	+	0.656191i	$0.227834\pi$
$151$			198.170i			1.31238i	−0.754595	+	0.656191i	$0.772166\pi$
$152$		0			0
$153$	2.15726			0.0140997
$154$		0			0
$155$	−110.120			−0.710454
$156$		0			0
$157$		−	10.6882i		−	0.0680775i	−0.999421	−	0.0340387i	$-0.989163\pi$
$157$		−	10.6882i		−	0.0680775i	0.999421	−	0.0340387i	$-0.0108370\pi$
$158$		0			0
$159$	167.919			1.05609
$160$		0			0
$161$	−133.304			−0.827973
$162$		0			0
$163$		−	16.4939i		−	0.101190i	−0.998719	−	0.0505948i	$-0.983888\pi$
$163$		−	16.4939i		−	0.101190i	0.998719	−	0.0505948i	$-0.0161117\pi$
$164$		0			0
$165$	159.388			0.965985
$166$		0			0
$167$		−	93.1887i		−	0.558016i	−0.960289	−	0.279008i	$-0.909994\pi$
$167$		−	93.1887i		−	0.558016i	0.960289	−	0.279008i	$-0.0900056\pi$
$168$		0			0
$169$	436.604			2.58346
$170$		0			0
$171$	−4.04329	−	12.2084i	−0.0236450	−	0.0713941i
$172$		0			0
$173$	−203.740			−1.17769			−0.588844	−	0.808247i	$-0.700417\pi$
$173$	−203.740			−1.17769			−0.588844	+	0.808247i	$0.700417\pi$
$174$		0			0
$175$	−77.9140			−0.445223
$176$		0			0
$177$	287.269			1.62299
$178$		0			0
$179$	134.059			0.748931			0.374466	−	0.927241i	$-0.377826\pi$
$179$	134.059			0.748931			0.374466	+	0.927241i	$0.377826\pi$
$180$		0			0
$181$	−133.672			−0.738517			−0.369259	−	0.929327i	$-0.620388\pi$
$181$	−133.672			−0.738517			−0.369259	+	0.929327i	$0.620388\pi$
$182$		0			0
$183$		−	159.060i		−	0.869180i
$184$		0			0
$185$			32.4940i			0.175643i
$186$		0			0
$187$		−	42.9188i		−	0.229512i
$188$		0			0
$189$	266.285			1.40891
$190$		0			0
$191$	217.180			1.13707			0.568533	−	0.822660i	$-0.307511\pi$
$191$	217.180			1.13707			0.568533	+	0.822660i	$0.307511\pi$
$192$		0			0
$193$			109.764i			0.568724i	0.958717	+	0.284362i	$0.0917817\pi$
$193$			109.764i			0.568724i	−0.958717	+	0.284362i	$0.908218\pi$
$194$		0			0
$195$		−	291.272i		−	1.49370i
$196$		0			0
$197$		−	355.978i		−	1.80700i	−0.428592	−	0.903498i	$-0.640990\pi$
$197$		−	355.978i		−	1.80700i	0.428592	−	0.903498i	$-0.359010\pi$
$198$		0			0
$199$	−198.819			−0.999089			−0.499545	−	0.866288i	$-0.666499\pi$
$199$	−198.819			−0.999089			−0.499545	+	0.866288i	$0.666499\pi$
$200$		0			0
$201$	173.428			0.862826
$202$		0			0
$203$	−359.305			−1.76998
$204$		0			0
$205$	100.103			0.488307
$206$		0			0
$207$	−9.45974			−0.0456992
$208$		0			0
$209$	−242.887	+	80.4416i	−1.16214	+	0.384888i
$210$		0			0
$211$	57.6385			0.273168			0.136584	−	0.990628i	$-0.456388\pi$
$211$	57.6385			0.273168			0.136584	+	0.990628i	$0.456388\pi$
$212$		0			0
$213$			190.099i			0.892486i
$214$		0			0
$215$	−181.348			−0.843481
$216$		0			0
$217$			256.021i			1.17982i
$218$		0			0
$219$	332.841			1.51982
$220$		0			0
$221$	−78.4316			−0.354894
$222$		0			0
$223$		−	262.831i		−	1.17862i	−0.807908	−	0.589308i	$-0.799400\pi$
$223$		−	262.831i		−	1.17862i	0.807908	−	0.589308i	$-0.200600\pi$
$224$		0			0
$225$	−5.52907			−0.0245737
$226$		0			0
$227$	−100.940			−0.444671			−0.222335	−	0.974970i	$-0.571368\pi$
$227$	−100.940			−0.444671			−0.222335	+	0.974970i	$0.571368\pi$
$228$		0			0
$229$			81.0767i			0.354047i	0.984207	+	0.177023i	$0.0566468\pi$
$229$			81.0767i			0.354047i	−0.984207	+	0.177023i	$0.943353\pi$
$230$		0			0
$231$		−	370.563i		−	1.60417i
$232$		0			0
$233$	−70.8655			−0.304144			−0.152072	−	0.988369i	$-0.548595\pi$
$233$	−70.8655			−0.304144			−0.152072	+	0.988369i	$0.548595\pi$
$234$		0			0
$235$			10.9607i			0.0466413i
$236$		0			0
$237$			256.908i			1.08400i
$238$		0			0
$239$	287.638			1.20351			0.601754	−	0.798682i	$-0.294469\pi$
$239$	287.638			1.20351			0.601754	+	0.798682i	$0.294469\pi$
$240$		0			0
$241$		−	253.309i		−	1.05107i	−0.850771	−	0.525537i	$-0.823864\pi$
$241$		−	253.309i		−	1.05107i	0.850771	−	0.525537i	$-0.176136\pi$
$242$		0			0
$243$	36.4714			0.150088
$244$		0			0
$245$		−	172.219i		−	0.702936i
$246$		0			0
$247$	147.002	+	443.862i	0.595151	+	1.79701i
$248$		0			0
$249$			84.0995i			0.337749i
$250$		0			0
$251$		−	121.701i		−	0.484866i	−0.970168	−	0.242433i	$-0.922054\pi$
$251$		−	121.701i		−	0.484866i	0.970168	−	0.242433i	$-0.0779455\pi$
$252$		0			0
$253$			188.202i			0.743883i
$254$		0			0
$255$			37.7225i			0.147931i
$256$		0			0
$257$			163.052i			0.634444i	0.948351	+	0.317222i	$0.102750\pi$
$257$			163.052i			0.634444i	−0.948351	+	0.317222i	$0.897250\pi$
$258$		0			0
$259$	75.5459			0.291683
$260$		0			0
$261$	−25.4977			−0.0976922
$262$		0			0
$263$	−112.306			−0.427017			−0.213509	−	0.976941i	$-0.568489\pi$
$263$	−112.306			−0.427017			−0.213509	+	0.976941i	$0.568489\pi$
$264$		0			0
$265$		−	238.790i		−	0.901094i
$266$		0			0
$267$		−	424.404i		−	1.58953i
$268$		0			0
$269$	180.426			0.670728			0.335364	−	0.942089i	$-0.391141\pi$
$269$	180.426			0.670728			0.335364	+	0.942089i	$0.391141\pi$
$270$		0			0
$271$	−96.8752			−0.357473			−0.178737	−	0.983897i	$-0.557201\pi$
$271$	−96.8752			−0.357473			−0.178737	+	0.983897i	$0.557201\pi$
$272$		0			0
$273$	−677.183			−2.48052
$274$		0			0
$275$			110.001i			0.400005i
$276$		0			0
$277$			156.836i			0.566195i	0.959091	+	0.283098i	$0.0913621\pi$
$277$			156.836i			0.566195i	−0.959091	+	0.283098i	$0.908638\pi$
$278$		0			0
$279$			18.1682i			0.0651191i
$280$		0			0
$281$			395.988i			1.40921i	0.709599	+	0.704606i	$0.248876\pi$
$281$			395.988i			1.40921i	−0.709599	+	0.704606i	$0.751124\pi$
$282$		0			0
$283$			160.652i			0.567676i	0.958872	+	0.283838i	$0.0916077\pi$
$283$			160.652i			0.567676i	−0.958872	+	0.283838i	$0.908392\pi$
$284$		0			0
$285$	213.480	−	70.7023i	0.749052	−	0.248078i
$286$		0			0
$287$		−	232.731i		−	0.810910i
$288$		0			0
$289$	−278.842			−0.964852
$290$		0			0
$291$			492.401i			1.69210i
$292$		0			0
$293$	−143.149			−0.488563			−0.244282	−	0.969704i	$-0.578552\pi$
$293$	−143.149			−0.488563			−0.244282	+	0.969704i	$0.578552\pi$
$294$		0			0
$295$		−	408.513i		−	1.38479i
$296$		0			0
$297$		−	375.949i		−	1.26582i
$298$		0			0
$299$	343.929			1.15026
$300$		0			0
$301$			421.621i			1.40073i
$302$		0			0
$303$		−	286.974i		−	0.947110i
$304$		0			0
$305$	−226.192			−0.741614
$306$		0			0
$307$	19.6254			0.0639265			0.0319632	−	0.999489i	$-0.489824\pi$
$307$	19.6254			0.0639265			0.0319632	+	0.999489i	$0.489824\pi$
$308$		0			0
$309$		−	134.521i		−	0.435342i
$310$		0			0
$311$	−47.9864			−0.154297			−0.0771485	−	0.997020i	$-0.524582\pi$
$311$	−47.9864			−0.154297			−0.0771485	+	0.997020i	$0.524582\pi$
$312$		0			0
$313$	42.4182			0.135521			0.0677607	−	0.997702i	$-0.478415\pi$
$313$	42.4182			0.135521			0.0677607	+	0.997702i	$0.478415\pi$
$314$		0			0
$315$		−	26.4870i		−	0.0840858i
$316$		0			0
$317$	−120.624			−0.380518			−0.190259	−	0.981734i	$-0.560933\pi$
$317$	−120.624			−0.380518			−0.190259	+	0.981734i	$0.560933\pi$
$318$		0			0
$319$			507.278i			1.59021i
$320$		0			0
$321$	−395.483			−1.23203
$322$		0			0
$323$	−19.0382	−	57.4844i	−0.0589419	−	0.177970i
$324$		0			0
$325$	201.021			0.618527
$326$		0			0
$327$	−229.819			−0.702811
$328$		0			0
$329$	25.4828			0.0774553
$330$		0			0
$331$	406.653			1.22856			0.614279	−	0.789089i	$-0.289447\pi$
$331$	406.653			1.22856			0.614279	+	0.789089i	$0.289447\pi$
$332$		0			0
$333$	5.36103			0.0160992
$334$		0			0
$335$		−	246.624i		−	0.736193i
$336$		0			0
$337$		−	483.585i		−	1.43497i	−0.696573	−	0.717486i	$-0.745293\pi$
$337$		−	483.585i		−	1.43497i	0.696573	−	0.717486i	$-0.254707\pi$
$338$		0			0
$339$			9.19983i			0.0271381i
$340$		0			0
$341$	361.458			1.05999
$342$		0			0
$343$	66.9776			0.195270
$344$		0			0
$345$		−	165.416i		−	0.479467i
$346$		0			0
$347$		−	610.149i		−	1.75836i	−0.476494	−	0.879178i	$-0.658092\pi$
$347$		−	610.149i		−	1.75836i	0.476494	−	0.879178i	$-0.341908\pi$
$348$		0			0
$349$			676.677i			1.93890i	0.245284	+	0.969451i	$0.421119\pi$
$349$			676.677i			1.93890i	−0.245284	+	0.969451i	$0.578881\pi$
$350$		0			0
$351$	−687.025			−1.95734
$352$		0			0
$353$	294.892			0.835389			0.417695	−	0.908587i	$-0.362838\pi$
$353$	294.892			0.835389			0.417695	+	0.908587i	$0.362838\pi$
$354$		0			0
$355$	270.332			0.761499
$356$		0			0
$357$	87.7017			0.245663
$358$		0			0
$359$	−335.771			−0.935294			−0.467647	−	0.883915i	$-0.654898\pi$
$359$	−335.771			−0.935294			−0.467647	+	0.883915i	$0.654898\pi$
$360$		0			0
$361$	−289.634	+	215.483i	−0.802311	+	0.596906i
$362$		0			0
$363$	−174.090			−0.479586
$364$		0			0
$365$		−	473.319i		−	1.29676i
$366$		0			0
$367$	−451.815			−1.23110			−0.615552	−	0.788096i	$-0.711067\pi$
$367$	−451.815			−1.23110			−0.615552	+	0.788096i	$0.711067\pi$
$368$		0			0
$369$		−	16.5155i		−	0.0447575i
$370$		0			0
$371$	−555.167			−1.49641
$372$		0			0
$373$	−401.339			−1.07598			−0.537988	−	0.842953i	$-0.680815\pi$
$373$	−401.339			−1.07598			−0.537988	+	0.842953i	$0.680815\pi$
$374$		0			0
$375$		−	392.582i		−	1.04689i
$376$		0			0
$377$	927.021			2.45894
$378$		0			0
$379$	−545.438			−1.43915			−0.719576	−	0.694414i	$-0.755664\pi$
$379$	−545.438			−1.43915			−0.719576	+	0.694414i	$0.755664\pi$
$380$		0			0
$381$		−	251.335i		−	0.659672i
$382$		0			0
$383$			393.116i			1.02641i	0.858266	+	0.513206i	$0.171542\pi$
$383$			393.116i			1.02641i	−0.858266	+	0.513206i	$0.828458\pi$
$384$		0			0
$385$	−526.962			−1.36873
$386$		0			0
$387$			29.9198i			0.0773121i
$388$		0			0
$389$			118.780i			0.305347i	0.988277	+	0.152673i	$0.0487883\pi$
$389$			118.780i			0.305347i	−0.988277	+	0.152673i	$0.951212\pi$
$390$		0			0
$391$	−44.5421			−0.113918
$392$		0			0
$393$			745.984i			1.89818i
$394$		0			0
$395$	365.338			0.924906
$396$		0			0
$397$			180.587i			0.454880i	0.973792	+	0.227440i	$0.0730356\pi$
$397$			180.587i			0.454880i	−0.973792	+	0.227440i	$0.926964\pi$
$398$		0			0
$399$	−164.377	−	496.324i	−0.411973	−	1.24392i
$400$		0			0
$401$		−	660.676i		−	1.64757i	−0.566901	−	0.823786i	$-0.691858\pi$
$401$		−	660.676i		−	1.64757i	0.566901	−	0.823786i	$-0.308142\pi$
$402$		0			0
$403$		−	660.544i		−	1.63907i
$404$		0			0
$405$			305.439i			0.754171i
$406$		0			0
$407$		−	106.658i		−	0.262059i
$408$		0			0
$409$		−	164.761i		−	0.402839i	−0.979505	−	0.201419i	$-0.935445\pi$
$409$		−	164.761i		−	0.402839i	0.979505	−	0.201419i	$-0.0645554\pi$
$410$		0			0
$411$	171.533			0.417356
$412$		0			0
$413$	−949.759			−2.29966
$414$		0			0
$415$	119.594			0.288179
$416$		0			0
$417$			590.099i			1.41510i
$418$		0			0
$419$		−	211.435i		−	0.504619i	−0.967647	−	0.252309i	$-0.918810\pi$
$419$		−	211.435i		−	0.504619i	0.967647	−	0.252309i	$-0.0811901\pi$
$420$		0			0
$421$	−300.457			−0.713675			−0.356837	−	0.934167i	$-0.616145\pi$
$421$	−300.457			−0.713675			−0.356837	+	0.934167i	$0.616145\pi$
$422$		0			0
$423$	1.80836			0.00427507
$424$		0			0
$425$	−26.0342			−0.0612569
$426$		0			0
$427$			525.879i			1.23157i
$428$		0			0
$429$			956.068i			2.22860i
$430$		0			0
$431$		−	137.886i		−	0.319922i	−0.987123	−	0.159961i	$-0.948863\pi$
$431$		−	137.886i		−	0.319922i	0.987123	−	0.159961i	$-0.0511369\pi$
$432$		0			0
$433$		−	30.0480i		−	0.0693950i	−0.999398	−	0.0346975i	$-0.988953\pi$
$433$		−	30.0480i		−	0.0693950i	0.999398	−	0.0346975i	$-0.0110468\pi$
$434$		0			0
$435$		−	445.860i		−	1.02497i
$436$		0			0
$437$	83.4841	+	252.074i	0.191039	+	0.576828i
$438$		0			0
$439$		−	82.6971i		−	0.188376i	−0.995554	−	0.0941881i	$-0.969975\pi$
$439$		−	82.6971i		−	0.188376i	0.995554	−	0.0941881i	$-0.0300255\pi$
$440$		0			0
$441$	−28.4136			−0.0644300
$442$		0			0
$443$		−	67.5669i		−	0.152521i	−0.997088	−	0.0762606i	$-0.975702\pi$
$443$		−	67.5669i		−	0.152521i	0.997088	−	0.0762606i	$-0.0242981\pi$
$444$		0			0
$445$	−603.527			−1.35624
$446$		0			0
$447$			678.092i			1.51698i
$448$		0			0
$449$			251.365i			0.559832i	0.960024	+	0.279916i	$0.0903066\pi$
$449$			251.365i			0.559832i	−0.960024	+	0.279916i	$0.909693\pi$
$450$		0			0
$451$	−328.577			−0.728553
$452$		0			0
$453$			571.717i			1.26207i
$454$		0			0
$455$			962.992i			2.11647i
$456$		0			0
$457$	557.314			1.21950			0.609752	−	0.792592i	$-0.291269\pi$
$457$	557.314			1.21950			0.609752	+	0.792592i	$0.291269\pi$
$458$		0			0
$459$	88.9764			0.193848
$460$		0			0
$461$			40.6993i			0.0882848i	0.999025	+	0.0441424i	$0.0140555\pi$
$461$			40.6993i			0.0882848i	−0.999025	+	0.0441424i	$0.985944\pi$
$462$		0			0
$463$	−460.527			−0.994658			−0.497329	−	0.867562i	$-0.665686\pi$
$463$	−460.527			−0.994658			−0.497329	+	0.867562i	$0.665686\pi$
$464$		0			0
$465$	−317.695			−0.683216
$466$		0			0
$467$		−	147.884i		−	0.316667i	−0.987386	−	0.158334i	$-0.949388\pi$
$467$		−	147.884i		−	0.316667i	0.987386	−	0.158334i	$-0.0506122\pi$
$468$		0			0
$469$	−573.382			−1.22256
$470$		0			0
$471$		−	30.8352i		−	0.0654674i
$472$		0			0
$473$	595.257			1.25847
$474$		0			0
$475$	48.7952	+	147.333i	0.102727	+	0.310175i
$476$		0			0
$477$	−39.3968			−0.0825929
$478$		0			0
$479$	291.482			0.608522			0.304261	−	0.952589i	$-0.401590\pi$
$479$	291.482			0.608522			0.304261	+	0.952589i	$0.401590\pi$
$480$		0			0
$481$	−194.911			−0.405221
$482$		0			0
$483$	−384.579			−0.796230
$484$		0			0
$485$	700.222			1.44376
$486$		0			0
$487$		−	319.006i		−	0.655044i	−0.944844	−	0.327522i	$-0.893787\pi$
$487$		−	319.006i		−	0.655044i	0.944844	−	0.327522i	$-0.106213\pi$
$488$		0			0
$489$		−	47.5846i		−	0.0973100i
$490$		0			0
$491$			349.737i			0.712295i	0.934430	+	0.356147i	$0.115910\pi$
$491$			349.737i			0.712295i	−0.934430	+	0.356147i	$0.884090\pi$
$492$		0			0
$493$	−120.058			−0.243526
$494$		0			0
$495$	−37.3952			−0.0755459
$496$		0			0
$497$		−	628.501i		−	1.26459i
$498$		0			0
$499$		−	511.589i		−	1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$499$		−	511.589i		−	1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$500$		0			0
$501$		−	268.848i		−	0.536623i
$502$		0			0
$503$	584.321			1.16167			0.580836	−	0.814021i	$-0.302726\pi$
$503$	584.321			1.16167			0.580836	+	0.814021i	$0.302726\pi$
$504$		0			0
$505$	−408.094			−0.808107
$506$		0			0
$507$	1259.60			2.48441
$508$		0			0
$509$	114.952			0.225839			0.112919	−	0.993604i	$-0.463980\pi$
$509$	114.952			0.225839			0.112919	+	0.993604i	$0.463980\pi$
$510$		0			0
$511$	−1100.43			−2.15348
$512$		0			0
$513$	−166.766	−	503.537i	−0.325080	−	0.981554i
$514$		0			0
$515$	−191.296			−0.371448
$516$		0			0
$517$		−	35.9774i		−	0.0695888i
$518$		0			0
$519$	−587.787			−1.13254
$520$		0			0
$521$		−	825.108i		−	1.58370i	−0.610715	−	0.791851i	$-0.709118\pi$
$521$		−	825.108i		−	1.58370i	0.610715	−	0.791851i	$-0.290882\pi$
$522$		0			0
$523$	−851.469			−1.62805			−0.814024	−	0.580832i	$-0.802727\pi$
$523$	−851.469			−1.62805			−0.814024	+	0.580832i	$0.802727\pi$
$524$		0			0
$525$	−224.781			−0.428153
$526$		0			0
$527$			85.5468i			0.162328i
$528$		0			0
$529$	−333.679			−0.630774
$530$		0			0
$531$	−67.3985			−0.126928
$532$		0			0
$533$			600.456i			1.12656i
$534$		0			0
$535$			562.399i			1.05121i
$536$		0			0
$537$	386.757			0.720218
$538$		0			0
$539$			565.292i			1.04878i
$540$		0			0
$541$		−	628.869i		−	1.16242i	−0.813754	−	0.581210i	$-0.802580\pi$
$541$		−	628.869i		−	1.16242i	0.813754	−	0.581210i	$-0.197420\pi$
$542$		0			0
$543$	−385.640			−0.710203
$544$		0			0
$545$			326.816i			0.599662i
$546$		0			0
$547$	−269.994			−0.493590			−0.246795	−	0.969068i	$-0.579377\pi$
$547$	−269.994			−0.493590			−0.246795	+	0.969068i	$0.579377\pi$
$548$		0			0
$549$			37.3184i			0.0679752i
$550$		0			0
$551$	225.022	+	679.436i	0.408389	+	1.23310i
$552$		0			0
$553$		−	849.381i		−	1.53595i
$554$		0			0
$555$			93.7446i			0.168909i
$556$		0			0
$557$		−	555.959i		−	0.998131i	−0.866564	−	0.499066i	$-0.833677\pi$
$557$		−	555.959i		−	0.998131i	0.866564	−	0.499066i	$-0.166323\pi$
$558$		0			0
$559$		−	1087.80i		−	1.94597i
$560$		0			0
$561$		−	123.820i		−	0.220713i
$562$		0			0
$563$	899.134			1.59704			0.798520	−	0.601968i	$-0.205617\pi$
$563$	899.134			1.59704			0.798520	+	0.601968i	$0.205617\pi$
$564$		0			0
$565$	13.0827			0.0231552
$566$		0			0
$567$	710.122			1.25242
$568$		0			0
$569$			790.678i			1.38959i	0.719206	+	0.694797i	$0.244506\pi$
$569$			790.678i			1.38959i	−0.719206	+	0.694797i	$0.755494\pi$
$570$		0			0
$571$		−	295.018i		−	0.516670i	−0.966055	−	0.258335i	$-0.916826\pi$
$571$		−	295.018i		−	0.516670i	0.966055	−	0.258335i	$-0.0831738\pi$
$572$		0			0
$573$	626.560			1.09347
$574$		0			0
$575$	114.162			0.198542
$576$		0			0
$577$	−939.760			−1.62870			−0.814350	−	0.580374i	$-0.802907\pi$
$577$	−939.760			−1.62870			−0.814350	+	0.580374i	$0.802907\pi$
$578$		0			0
$579$			316.667i			0.546920i
$580$		0			0
$581$		−	278.047i		−	0.478566i
$582$		0			0
$583$			783.803i			1.34443i
$584$		0			0
$585$			68.3376i			0.116816i
$586$		0			0
$587$		−	93.0053i		−	0.158442i	−0.996857	−	0.0792209i	$-0.974757\pi$
$587$		−	93.0053i		−	0.158442i	0.996857	−	0.0792209i	$-0.0252432\pi$
$588$		0			0
$589$	484.129	−	160.338i	0.821950	−	0.272221i
$590$		0			0
$591$		−	1026.99i		−	1.73772i
$592$		0			0
$593$	704.015			1.18721			0.593605	−	0.804757i	$-0.297704\pi$
$593$	704.015			1.18721			0.593605	+	0.804757i	$0.297704\pi$
$594$		0			0
$595$		−	124.717i		−	0.209608i
$596$		0			0
$597$	−573.589			−0.960785
$598$		0			0
$599$		−	1176.63i		−	1.96433i	−0.188017	−	0.982166i	$-0.560206\pi$
$599$		−	1176.63i		−	1.96433i	0.188017	−	0.982166i	$-0.439794\pi$
$600$		0			0
$601$			520.928i			0.866769i	0.901209	+	0.433385i	$0.142681\pi$
$601$			520.928i			0.866769i	−0.901209	+	0.433385i	$0.857319\pi$
$602$		0			0
$603$	−40.6894			−0.0674782
$604$		0			0
$605$			247.566i			0.409199i
$606$		0			0
$607$			12.8069i			0.0210987i	0.999944	+	0.0105494i	$0.00335803\pi$
$607$			12.8069i			0.0210987i	−0.999944	+	0.0105494i	$0.996642\pi$
$608$		0			0
$609$	−1036.59			−1.70212
$610$		0			0
$611$	−65.7466			−0.107605
$612$		0			0
$613$		−	833.193i		−	1.35920i	−0.733581	−	0.679602i	$-0.762152\pi$
$613$		−	833.193i		−	1.35920i	0.733581	−	0.679602i	$-0.237848\pi$
$614$		0			0
$615$	288.795			0.469586
$616$		0			0
$617$	101.566			0.164612			0.0823061	−	0.996607i	$-0.473771\pi$
$617$	101.566			0.164612			0.0823061	+	0.996607i	$0.473771\pi$
$618$		0			0
$619$		−	941.352i		−	1.52076i	−0.649477	−	0.760381i	$-0.725012\pi$
$619$		−	941.352i		−	1.52076i	0.649477	−	0.760381i	$-0.274988\pi$
$620$		0			0
$621$	−390.169			−0.628291
$622$		0			0
$623$			1403.15i			2.25225i
$624$		0			0
$625$	−354.059			−0.566495
$626$		0			0
$627$	−700.725	+	232.073i	−1.11758	+	0.370132i
$628$		0			0
$629$	25.2429			0.0401318
$630$		0			0
$631$	−748.857			−1.18678			−0.593389	−	0.804916i	$-0.702211\pi$
$631$	−748.857			−1.18678			−0.593389	+	0.804916i	$0.702211\pi$
$632$		0			0
$633$	166.286			0.262695
$634$		0			0
$635$	−357.413			−0.562855
$636$		0			0
$637$	1033.04			1.62172
$638$		0			0
$639$		−	44.6008i		−	0.0697978i
$640$		0			0
$641$			140.831i			0.219705i	0.993948	+	0.109853i	$0.0350379\pi$
$641$			140.831i			0.219705i	−0.993948	+	0.109853i	$0.964962\pi$
$642$		0			0
$643$		−	13.7192i		−	0.0213363i	−0.999943	−	0.0106681i	$-0.996604\pi$
$643$		−	13.7192i		−	0.0213363i	0.999943	−	0.0106681i	$-0.00339584\pi$
$644$		0			0
$645$	−523.187			−0.811143
$646$		0			0
$647$	−356.048			−0.550305			−0.275153	−	0.961401i	$-0.588728\pi$
$647$	−356.048			−0.550305			−0.275153	+	0.961401i	$0.588728\pi$
$648$		0			0
$649$			1340.90i			2.06610i
$650$		0			0
$651$			738.616i			1.13459i
$652$		0			0
$653$		−	366.647i		−	0.561482i	−0.959784	−	0.280741i	$-0.909420\pi$
$653$		−	366.647i		−	0.561482i	0.959784	−	0.280741i	$-0.0905801\pi$
$654$		0			0
$655$	1060.83			1.61959
$656$		0			0
$657$	−78.0906			−0.118859
$658$		0			0
$659$	−526.050			−0.798255			−0.399127	−	0.916895i	$-0.630687\pi$
$659$	−526.050			−0.798255			−0.399127	+	0.916895i	$0.630687\pi$
$660$		0			0
$661$	1126.14			1.70370			0.851849	−	0.523787i	$-0.175481\pi$
$661$	1126.14			1.70370			0.851849	+	0.523787i	$0.175481\pi$
$662$		0			0
$663$	−226.274			−0.341288
$664$		0			0
$665$	−705.800	+	233.754i	−1.06135	+	0.351509i
$666$		0			0
$667$	526.465			0.789302
$668$		0			0
$669$		−	758.264i		−	1.13343i
$670$		0			0
$671$	742.452			1.10649
$672$		0			0
$673$		−	532.487i		−	0.791214i	−0.918420	−	0.395607i	$-0.870534\pi$
$673$		−	532.487i		−	0.791214i	0.918420	−	0.395607i	$-0.129466\pi$
$674$		0			0
$675$	−228.048			−0.337848
$676$		0			0
$677$	−1188.13			−1.75499			−0.877495	−	0.479585i	$-0.840787\pi$
$677$	−1188.13			−1.75499			−0.877495	+	0.479585i	$0.840787\pi$
$678$		0			0
$679$		−	1627.96i		−	2.39758i
$680$		0			0
$681$	−291.211			−0.427622
$682$		0			0
$683$	514.447			0.753216			0.376608	−	0.926373i	$-0.377090\pi$
$683$	514.447			0.753216			0.376608	+	0.926373i	$0.377090\pi$
$684$		0			0
$685$		−	243.930i		−	0.356102i
$686$		0			0
$687$			233.905i			0.340473i
$688$		0			0
$689$	1432.35			2.07889
$690$		0			0
$691$			58.6469i			0.0848725i	0.999099	+	0.0424363i	$0.0135119\pi$
$691$			58.6469i			0.0848725i	−0.999099	+	0.0424363i	$0.986488\pi$
$692$		0			0
$693$			86.9409i			0.125456i
$694$		0			0
$695$	839.154			1.20742
$696$		0			0
$697$		−	77.7648i		−	0.111571i
$698$		0			0
$699$	−204.446			−0.292483
$700$		0			0
$701$		−	455.578i		−	0.649898i	−0.945732	−	0.324949i	$-0.894653\pi$
$701$		−	455.578i		−	0.649898i	0.945732	−	0.324949i	$-0.105347\pi$
$702$		0			0
$703$	−47.3122	−	142.855i	−0.0673004	−	0.203208i
$704$		0			0
$705$			31.6215i			0.0448532i
$706$		0			0
$707$			948.785i			1.34199i
$708$		0			0
$709$		−	273.301i		−	0.385474i	−0.981250	−	0.192737i	$-0.938264\pi$
$709$		−	273.301i		−	0.385474i	0.981250	−	0.192737i	$-0.0617365\pi$
$710$		0			0
$711$		−	60.2753i		−	0.0847754i
$712$		0			0
$713$		−	375.130i		−	0.526128i
$714$		0			0
$715$	1359.58			1.90151
$716$		0			0
$717$	829.832			1.15737
$718$		0			0
$719$	1176.79			1.63671			0.818353	−	0.574716i	$-0.194888\pi$
$719$	1176.79			1.63671			0.818353	+	0.574716i	$0.194888\pi$
$720$		0			0
$721$			444.748i			0.616848i
$722$		0			0
$723$		−	730.792i		−	1.01078i
$724$		0			0
$725$	307.711			0.424428
$726$		0			0
$727$	199.545			0.274477			0.137238	−	0.990538i	$-0.456177\pi$
$727$	199.545			0.274477			0.137238	+	0.990538i	$0.456177\pi$
$728$		0			0
$729$	775.270			1.06347
$730$		0			0
$731$			140.880i			0.192723i
$732$		0			0
$733$			1361.53i			1.85747i	0.370739	+	0.928737i	$0.379104\pi$
$733$			1361.53i			1.85747i	−0.370739	+	0.928737i	$0.620896\pi$
$734$		0			0
$735$		−	496.850i		−	0.675986i
$736$		0			0
$737$			809.519i			1.09840i
$738$		0			0
$739$			220.790i			0.298769i	0.988779	+	0.149384i	$0.0477292\pi$
$739$			220.790i			0.298769i	−0.988779	+	0.149384i	$0.952271\pi$
$740$		0			0
$741$	424.100	+	1280.53i	0.572334	+	1.72812i
$742$		0			0
$743$			564.337i			0.759538i	0.925081	+	0.379769i	$0.123997\pi$
$743$			564.337i			0.759538i	−0.925081	+	0.379769i	$0.876003\pi$
$744$		0			0
$745$	964.285			1.29434
$746$		0			0
$747$		−	19.7313i		−	0.0264140i
$748$		0			0
$749$	1307.53			1.74570
$750$		0			0
$751$			1210.91i			1.61240i	0.591642	+	0.806201i	$0.298480\pi$
$751$			1210.91i			1.61240i	−0.591642	+	0.806201i	$0.701520\pi$
$752$		0			0
$753$		−	351.107i		−	0.466277i
$754$		0			0
$755$	813.013			1.07684
$756$		0			0
$757$		−	630.294i		−	0.832620i	−0.909223	−	0.416310i	$-0.863323\pi$
$757$		−	630.294i		−	0.832620i	0.909223	−	0.416310i	$-0.136677\pi$
$758$		0			0
$759$			542.961i			0.715363i
$760$		0			0
$761$	1419.02			1.86468			0.932342	−	0.361577i	$-0.117762\pi$
$761$	1419.02			1.86468			0.932342	+	0.361577i	$0.117762\pi$
$762$		0			0
$763$	759.821			0.995834
$764$		0			0
$765$		−	8.85038i		−	0.0115691i
$766$		0			0
$767$	2450.42			3.19480
$768$		0			0
$769$	−168.690			−0.219363			−0.109682	−	0.993967i	$-0.534983\pi$
$769$	−168.690			−0.219363			−0.109682	+	0.993967i	$0.534983\pi$
$770$		0			0
$771$			470.403i			0.610120i
$772$		0			0
$773$	1042.83			1.34907			0.674533	−	0.738244i	$-0.264345\pi$
$773$	1042.83			1.34907			0.674533	+	0.738244i	$0.264345\pi$
$774$		0			0
$775$		−	219.258i		−	0.282913i
$776$		0			0
$777$	217.949			0.280500
$778$		0			0
$779$	−440.089	+	145.753i	−0.564940	+	0.187102i
$780$		0			0
$781$	−887.337			−1.13615
$782$		0			0
$783$	−1051.66			−1.34311
$784$		0			0
$785$	−43.8494			−0.0558590
$786$		0			0
$787$	578.516			0.735090			0.367545	−	0.930006i	$-0.380198\pi$
$787$	578.516			0.735090			0.367545	+	0.930006i	$0.380198\pi$
$788$		0			0
$789$	−324.000			−0.410646
$790$		0			0
$791$		−	30.4162i		−	0.0384528i
$792$		0			0
$793$		−	1356.79i		−	1.71096i
$794$		0			0
$795$		−	688.905i		−	0.866547i
$796$		0			0
$797$	1207.33			1.51484			0.757421	−	0.652926i	$-0.226459\pi$
$797$	1207.33			1.51484			0.757421	+	0.652926i	$0.226459\pi$
$798$		0			0
$799$	8.51482			0.0106568
$800$		0			0
$801$			99.5730i			0.124311i
$802$		0			0
$803$			1553.62i			1.93477i
$804$		0			0
$805$			546.893i			0.679370i
$806$		0			0
$807$	520.526			0.645013
$808$		0			0
$809$	−201.797			−0.249440			−0.124720	−	0.992192i	$-0.539803\pi$
$809$	−201.797			−0.249440			−0.124720	+	0.992192i	$0.539803\pi$
$810$		0			0
$811$	−938.736			−1.15750			−0.578752	−	0.815503i	$-0.696460\pi$
$811$	−938.736			−1.15750			−0.578752	+	0.815503i	$0.696460\pi$
$812$		0			0
$813$	−279.483			−0.343768
$814$		0			0
$815$	−67.6680			−0.0830282
$816$		0			0
$817$	797.273	−	264.048i	0.975854	−	0.323193i
$818$		0			0
$819$	158.879			0.193992
$820$		0			0
$821$		−	458.588i		−	0.558573i	−0.960208	−	0.279286i	$-0.909902\pi$
$821$		−	458.588i		−	0.558573i	0.960208	−	0.279286i	$-0.0900979\pi$
$822$		0			0
$823$	−487.247			−0.592037			−0.296019	−	0.955182i	$-0.595659\pi$
$823$	−487.247			−0.592037			−0.296019	+	0.955182i	$0.595659\pi$
$824$		0			0
$825$			317.352i			0.384669i
$826$		0			0
$827$	−1185.67			−1.43370			−0.716849	−	0.697228i	$-0.754416\pi$
$827$	−1185.67			−1.43370			−0.716849	+	0.697228i	$0.754416\pi$
$828$		0			0
$829$	−494.899			−0.596983			−0.298492	−	0.954412i	$-0.596483\pi$
$829$	−494.899			−0.596983			−0.298492	+	0.954412i	$0.596483\pi$
$830$		0			0
$831$			452.470i			0.544488i
$832$		0			0
$833$	−133.788			−0.160610
$834$		0			0
$835$	−382.317			−0.457865
$836$		0			0
$837$			749.351i			0.895282i
$838$		0			0
$839$		−	877.317i		−	1.04567i	−0.852434	−	0.522835i	$-0.824874\pi$
$839$		−	877.317i		−	1.04567i	0.852434	−	0.522835i	$-0.175126\pi$
$840$		0			0
$841$	578.026			0.687308
$842$		0			0
$843$			1142.42i			1.35518i
$844$		0			0
$845$		−	1791.22i		−	2.11978i
$846$		0			0
$847$	575.570			0.679540
$848$		0			0
$849$			463.479i			0.545912i
$850$		0			0
$851$	−110.692			−0.130073
$852$		0			0
$853$		−	611.241i		−	0.716578i	−0.933611	−	0.358289i	$-0.883360\pi$
$853$		−	611.241i		−	0.716578i	0.933611	−	0.358289i	$-0.116640\pi$
$854$		0			0
$855$	−50.0863	+	16.5880i	−0.0585804	+	0.0194012i
$856$		0			0
$857$			1392.09i			1.62438i	0.583392	+	0.812191i	$0.301725\pi$
$857$			1392.09i			1.62438i	−0.583392	+	0.812191i	$0.698275\pi$
$858$		0			0
$859$		−	583.617i		−	0.679415i	−0.940531	−	0.339707i	$-0.889672\pi$
$859$		−	583.617i		−	0.679415i	0.940531	−	0.339707i	$-0.110328\pi$
$860$		0			0
$861$		−	671.426i		−	0.779821i
$862$		0			0
$863$		−	999.854i		−	1.15858i	−0.815122	−	0.579290i	$-0.803330\pi$
$863$		−	999.854i		−	1.15858i	0.815122	−	0.579290i	$-0.196670\pi$
$864$		0			0
$865$			835.866i			0.966319i
$866$		0			0
$867$	−804.456			−0.927861
$868$		0			0
$869$	−1199.18			−1.37996
$870$		0			0
$871$	1479.35			1.69845
$872$		0			0
$873$		−	115.526i		−	0.132332i
$874$		0			0
$875$			1297.94i			1.48336i
$876$		0			0
$877$	−706.258			−0.805311			−0.402656	−	0.915352i	$-0.631913\pi$
$877$	−706.258			−0.805311			−0.402656	+	0.915352i	$0.631913\pi$
$878$		0			0
$879$	−412.983			−0.469832
$880$		0			0
$881$	−208.610			−0.236788			−0.118394	−	0.992967i	$-0.537775\pi$
$881$	−208.610			−0.236788			−0.118394	+	0.992967i	$0.537775\pi$
$882$		0			0
$883$		−	485.534i		−	0.549869i	−0.961463	−	0.274934i	$-0.911344\pi$
$883$		−	485.534i		−	0.549869i	0.961463	−	0.274934i	$-0.0886562\pi$
$884$		0			0
$885$		−	1178.55i		−	1.33170i
$886$		0			0
$887$		−	375.943i		−	0.423836i	−0.977287	−	0.211918i	$-0.932029\pi$
$887$		−	375.943i		−	0.423836i	0.977287	−	0.211918i	$-0.0679710\pi$
$888$		0			0
$889$			830.956i			0.934709i
$890$		0			0
$891$		−	1002.57i		−	1.12522i
$892$		0			0
$893$	−15.9591	−	48.1873i	−0.0178713	−	0.0539611i
$894$		0			0
$895$		−	549.991i		−	0.614515i
$896$		0			0
$897$	992.229			1.10616
$898$		0			0
$899$		−	1011.12i		−	1.12472i
$900$		0			0
$901$	−185.504			−0.205886
$902$		0			0
$903$			1216.37i			1.34703i
$904$		0			0
$905$			548.402i			0.605970i
$906$		0			0
$907$	−684.806			−0.755023			−0.377511	−	0.926005i	$-0.623220\pi$
$907$	−684.806			−0.755023			−0.377511	+	0.926005i	$0.623220\pi$
$908$		0			0
$909$			67.3294i			0.0740698i
$910$		0			0
$911$			1359.16i			1.49194i	0.665978	+	0.745971i	$0.268014\pi$
$911$			1359.16i			1.49194i	−0.665978	+	0.745971i	$0.731986\pi$
$912$		0			0
$913$	−392.555			−0.429962
$914$		0			0
$915$	−652.561			−0.713182
$916$		0			0
$917$		−	2466.35i		−	2.68958i
$918$		0			0
$919$	−1188.09			−1.29281			−0.646405	−	0.762994i	$-0.723728\pi$
$919$	−1188.09			−1.29281			−0.646405	+	0.762994i	$0.723728\pi$
$920$		0			0
$921$	56.6191			0.0614756
$922$		0			0
$923$			1621.56i			1.75683i
$924$		0			0
$925$	−64.6979			−0.0699437
$926$		0			0
$927$			31.5610i			0.0340464i
$928$		0			0
$929$	−110.462			−0.118904			−0.0594520	−	0.998231i	$-0.518935\pi$
$929$	−110.462			−0.118904			−0.0594520	+	0.998231i	$0.518935\pi$
$930$		0			0
$931$	250.756	+	757.138i	0.269341	+	0.813252i
$932$		0			0
$933$	−138.440			−0.148381
$934$		0			0
$935$	−176.079			−0.188320
$936$		0			0
$937$	−1090.40			−1.16371			−0.581855	−	0.813293i	$-0.697673\pi$
$937$	−1090.40			−1.16371			−0.581855	+	0.813293i	$0.697673\pi$
$938$		0			0
$939$	122.376			0.130326
$940$		0			0
$941$	−690.663			−0.733967			−0.366984	−	0.930227i	$-0.619609\pi$
$941$	−690.663			−0.733967			−0.366984	+	0.930227i	$0.619609\pi$
$942$		0			0
$943$			341.005i			0.361617i
$944$		0			0
$945$		−	1092.46i		−	1.15605i
$946$		0			0
$947$			1664.77i			1.75794i	0.476874	+	0.878972i	$0.341770\pi$
$947$			1664.77i			1.75794i	−0.476874	+	0.878972i	$0.658230\pi$
$948$		0			0
$949$	2839.15			2.99173
$950$		0			0
$951$	−347.999			−0.365930
$952$		0			0
$953$		−	331.839i		−	0.348204i	−0.984728	−	0.174102i	$-0.944298\pi$
$953$		−	331.839i		−	0.348204i	0.984728	−	0.174102i	$-0.0557023\pi$
$954$		0			0
$955$		−	891.004i		−	0.932988i
$956$		0			0
$957$			1463.49i			1.52925i
$958$		0			0
$959$	−567.117			−0.591363
$960$		0			0
$961$	240.532			0.250294
$962$		0			0
$963$	92.7874			0.0963525
$964$		0			0
$965$	450.318			0.466650
$966$		0			0
$967$	81.7661			0.0845565			0.0422782	−	0.999106i	$-0.486538\pi$
$967$	81.7661			0.0845565			0.0422782	+	0.999106i	$0.486538\pi$
$968$		0			0
$969$	−54.9250	−	165.842i	−0.0566821	−	0.171147i
$970$		0			0
$971$	−656.760			−0.676375			−0.338187	−	0.941079i	$-0.609814\pi$
$971$	−656.760			−0.676375			−0.338187	+	0.941079i	$0.609814\pi$
$972$		0			0
$973$		−	1950.96i		−	2.00510i
$974$		0			0
$975$	579.943			0.594813
$976$		0			0
$977$			541.671i			0.554422i	0.960809	+	0.277211i	$0.0894102\pi$
$977$			541.671i			0.554422i	−0.960809	+	0.277211i	$0.910590\pi$
$978$		0			0
$979$	1981.01			2.02351
$980$		0			0
$981$	53.9198			0.0549641
$982$		0			0
$983$		−	857.643i		−	0.872475i	−0.899832	−	0.436237i	$-0.856311\pi$
$983$		−	857.643i		−	0.872475i	0.899832	−	0.436237i	$-0.143689\pi$
$984$		0			0
$985$	−1460.44			−1.48268
$986$		0			0
$987$	73.5174			0.0744857
$988$		0			0
$989$		−	617.771i		−	0.624642i
$990$		0			0
$991$			203.753i			0.205603i	0.994702	+	0.102802i	$0.0327806\pi$
$991$			203.753i			0.205603i	−0.994702	+	0.102802i	$0.967219\pi$
$992$		0			0
$993$	1173.19			1.18146
$994$		0			0
$995$			815.676i			0.819775i
$996$		0			0
$997$			1270.79i			1.27461i	0.770610	+	0.637306i	$0.219951\pi$
$997$			1270.79i			1.27461i	−0.770610	+	0.637306i	$0.780049\pi$
$998$		0			0
$999$	221.116			0.221338

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.g.e.417.35	yes	48
4.3	odd	2	inner	1216.3.g.e.417.15	yes	48
8.3	odd	2	inner	1216.3.g.e.417.36	yes	48
8.5	even	2	inner	1216.3.g.e.417.16	yes	48
19.18	odd	2	inner	1216.3.g.e.417.13	✓	48
76.75	even	2	inner	1216.3.g.e.417.33	yes	48
152.37	odd	2	inner	1216.3.g.e.417.34	yes	48
152.75	even	2	inner	1216.3.g.e.417.14	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.3.g.e.417.13	✓	48	19.18	odd	2	inner
1216.3.g.e.417.14	yes	48	152.75	even	2	inner
1216.3.g.e.417.15	yes	48	4.3	odd	2	inner
1216.3.g.e.417.16	yes	48	8.5	even	2	inner
1216.3.g.e.417.33	yes	48	76.75	even	2	inner
1216.3.g.e.417.34	yes	48	152.37	odd	2	inner
1216.3.g.e.417.35	yes	48	1.1	even	1	trivial
1216.3.g.e.417.36	yes	48	8.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.88498 q^{3} -4.10261i q^{5} -9.53824 q^{7} -0.676870 q^{9} +O(q^{10})\)	\(q+2.88498 q^{3} -4.10261i q^{5} -9.53824 q^{7} -0.676870 q^{9} +13.4664i q^{11} +24.6090 q^{13} -11.8360i q^{15} -3.18711 q^{17} +(5.97352 + 18.0365i) q^{19} -27.5177 q^{21} +13.9757 q^{23} +8.16859 q^{25} -27.9176 q^{27} +37.6700 q^{29} -26.8415i q^{31} +38.8503i q^{33} +39.1317i q^{35} -7.92032 q^{37} +70.9966 q^{39} +24.3998i q^{41} -44.2032i q^{43} +2.77693i q^{45} -2.67164 q^{47} +41.9780 q^{49} -9.19475 q^{51} +58.2044 q^{53} +55.2473 q^{55} +(17.2335 + 52.0351i) q^{57} +99.5738 q^{59} -55.1338i q^{61} +6.45614 q^{63} -100.961i q^{65} +60.1141 q^{67} +40.3197 q^{69} +65.8927i q^{71} +115.370 q^{73} +23.5663 q^{75} -128.446i q^{77} +89.0501i q^{79} -74.4500 q^{81} +29.1508i q^{83} +13.0755i q^{85} +108.677 q^{87} -147.108i q^{89} -234.727 q^{91} -77.4374i q^{93} +(73.9969 - 24.5070i) q^{95} +170.677i q^{97} -9.11499i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 264 q^{9}+O(q^{10}) \) 48 * q + 264 * q^9	\( 48 q + 264 q^{9} + 24 q^{17} - 160 q^{25} + 328 q^{49} + 8 q^{57} - 472 q^{73} - 736 q^{81}+O(q^{100}) \) 48 * q + 264 * q^9 + 24 * q^17 - 160 * q^25 + 328 * q^49 + 8 * q^57 - 472 * q^73 - 736 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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